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Experimental design and statistical analyses of data

Experimental design and statistical analyses of data. Lesson 4: Analysis of variance II A posteriori tests Model control How to choose the best model. Growth of bean plants in four different media. Completely randomized design (one-way anova). How to do it with SAS. DATA medium;

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Experimental design and statistical analyses of data

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  1. Experimental design and statistical analyses of data Lesson 4: Analysis of variance II A posteriori tests Model control How to choose the best model

  2. Growth of bean plants in four different media Completely randomized design (one-way anova)

  3. How to do it with SAS

  4. DATA medium; /* 20 bean plants exposed to 4 different treatments (5 plants per treatment) Mn = extra mangan added to the soil Zn = ekstra zink added to the soil Cu = ekstra cupper added to the soil K = control soil The dependent variable (Mass) is the biomass of the plants at harvest */ INPUT treat $ mass ; /* treat = treatment */ /* mass = biomass of a plant */ CARDS; zn 61.7 zn 59.4 zn 60.5 zn 59.2 zn 57.6 cu 57.0 cu 58.4 cu 57.3 cu 57.8 cu 59.9 mn 62.3 mn 66.2 mn 65.2 mn 63.7 mn 64.1 k 58.1 k 56.3 k 58.9 k 57.4 k 56.1 ;

  5. PROC SORT;/* sort the observations according to treatment */ BY treat; RUN; /* compute average and 95% confidence limits for each treatment */ PROC MEANS N MEAN CLM; BY treat; RUN;

  6. 1 14:09 Wednesday, November 7, 2001   Analysis Variable : MASS ------------------------------------- TREAT=cu --------------------------------- N Mean Lower 95.0% CLM Upper 95.0% CLM -------------------------------------------------- 5 58.0800000 56.6550587 59.5049413 -------------------------------------------------- -------------------------------------- TREAT=k --------------------------------- N Mean Lower 95.0% CLM Upper 95.0% CLM -------------------------------------------------- 5 57.3600000 55.8866517 58.8333483 -------------------------------------------------- ------------------------------------- TREAT=mn --------------------------------- N Mean Lower 95.0% CLM Upper 95.0% CLM -------------------------------------------------- 5 64.3000000 62.4562230 66.1437770 -------------------------------------------------- ------------------------------------- TREAT=zn --------------------------------- N Mean Lower 95.0% CLM Upper 95.0% CLM -------------------------------------------------- 5 59.6800000 57.7777805 61.5822195 --------------------------------------------------

  7. PROC GLM; CLASS treat; MODEL mass = treat /SOLUTION; /* SOLUTION gives the estimated parameter values */ RUN;

  8. Class Levels Values TREAT 4 cu k mn zn Number of observations in data set = 20 General Linear Models Procedure Dependent Variable: MASS Sum of Mean Source DF Squares Square F Value Pr > F Model 3 145.82150 48.60717 26.72 0.0001 Error 16 29.10800 1.81925 Corrected Total 19 174.92950 R-Square C.V. Root MSE MASS Mean 0.833602 2.253439 1.3488 59.855 Source DF Type I SS Mean Square F Value Pr > F TREAT 3 145.82150 48.60717 26.72 0.0001 Source DF Type III SS Mean Square F Value Pr > F TREAT 3 145.82150 48.60717 26.72 0.0001

  9. T for H0: Pr > |T| Std Error of Parameter Estimate Parameter=0 Estimate INTERCEPT 59.68000000 B 98.94 0.0001 0.60319980 TREAT cu -1.60000000 B -1.88 0.0791 0.85305334 k -2.32000000 B -2.72 0.0151 0.85305334 mn 4.62000000 B 5.42 0.0001 0.85305334 zn 0.00000000 B . . . NOTE: The X'X matrix has been found to be singular and a generalized inverse was used to solve the normal equations. Estimates followed by the letter 'B' are biased, and are not unique estimators of the parameters.

  10. PROC GLM; CLASS treat; MODEL mass = treat /SOLUTION; /* SOLUTION gives the estimated parameter values */ /*Test for pairwise differences between treatmentsby linear contrasts */ CONTRAST 'Cu vs K' Treat 1 -1 0 0; CONTRAST 'Cu vs Mn' Treat 1 0 -1 0; CONTRAST 'Cu vs Zn' Treat 1 0 0 -1; CONTRAST 'K vs Mn' Treat 0 1 -1 0; CONTRAST 'K vs Zn' Treat 0 1 0 -1; CONTRAST 'Mn vs Zn' Treat 0 0 1 -1; /* test for whether the 3 treatments with added minerals aredifferent from the control */ CONTRAST 'K vs Cu, Mn Zn' Treat 1 -3 1 1; RUN;

  11. Contrast DF Contrast SS Mean Square F Value Pr > F Cu vs K 1 1.29600 1.29600 0.71 0.4111 Cu vs Mn 1 96.72100 96.72100 53.17 0.0001 Cu vs Zn 1 6.40000 6.40000 3.52 0.0791 K vs Mn 1 120.40900 120.40900 66.19 0.0001 K vs Zn 1 13.45600 13.45600 7.40 0.0151 Mn vs Zn 1 53.36100 53.36100 29.33 0.0001 K vs Cu, Mn Zn 1 41.50017 41.50017 22.81 0.0002

  12. PROC GLM; CLASS treat; MODEL mass = treat /SOLUTION; /* SOLUTION gives the estimated parameter values */ /* Test for differences between levels of treatment */ MEANS treat / BON DUNCAN SCHEFFE TUKEY DUNNETT('k'); RUN;

  13. Tukey's Studentized Range (HSD) Test for variable: MASS NOTE: This test controls the type I experimentwise error rate. Alpha= 0.05 Confidence= 0.95 df= 16 MSE= 1.81925 Critical Value of Studentized Range= 4.046 Minimum Significant Difference= 2.4406 Comparisons significant at the 0.05 level are indicated by '***'. Simultaneous Simultaneous Lower Difference Upper TREAT Confidence Between Confidence Comparison Limit Means Limit mn - zn 2.1794 4.6200 7.0606 *** mn - cu 3.7794 6.2200 8.6606 *** mn - k 4.4994 6.9400 9.3806 *** zn - mn -7.0606 -4.6200 -2.1794 *** zn - cu -0.8406 1.6000 4.0406 zn - k -0.1206 2.3200 4.7606 cu - mn -8.6606 -6.2200 -3.7794 *** cu - zn -4.0406 -1.6000 0.8406 cu - k -1.7206 0.7200 3.1606 k - mn -9.3806 -6.9400 -4.4994 *** k - zn -4.7606 -2.3200 0.1206 k - cu -3.1606 -0.7200 1.7206

  14. Bonferroni (Dunn) T tests for variable: MASS NOTE: This test controls the type I experimentwise error rate but generally has a higher type II error rate than Tukey's for all pairwise comparisons. Alpha= 0.05 Confidence= 0.95 df= 16 MSE= 1.81925 Critical Value of T= 3.00833 Minimum Significant Difference= 2.5663 Comparisons significant at the 0.05 level are indicated by '***'. Simultaneous Simultaneous Lower Difference Upper TREAT Confidence Between Confidence Comparison Limit Means Limit mn - zn 2.0537 4.6200 7.1863 *** mn - cu 3.6537 6.2200 8.7863 *** mn - k 4.3737 6.9400 9.5063 *** zn - mn -7.1863 -4.6200 -2.0537 *** zn - cu -0.9663 1.6000 4.1663 zn - k -0.2463 2.3200 4.8863 cu - mn -8.7863 -6.2200 -3.6537 *** cu - zn -4.1663 -1.6000 0.9663 cu - k -1.8463 0.7200 3.2863 k - mn -9.5063 -6.9400 -4.3737 *** k - zn -4.8863 -2.3200 0.2463 k - cu -3.2863 -0.7200 1.8463

  15. Scheffe's test for variable: MASS NOTE: This test controls the type I experimentwise error rate but generally has a higher type II error rate than Tukey's for all pairwise comparisons. Alpha= 0.05 Confidence= 0.95 df= 16 MSE= 1.81925 Critical Value of F= 3.23887 Minimum Significant Difference= 2.6591 Comparisons significant at the 0.05 level are indicated by '***'. Simultaneous Simultaneous Lower Difference Upper TREAT Confidence Between Confidence Comparison Limit Means Limit mn - zn 1.9609 4.6200 7.2791 *** mn - cu 3.5609 6.2200 8.8791 *** mn - k 4.2809 6.9400 9.5991 *** zn - mn -7.2791 -4.6200 -1.9609 *** zn - cu -1.0591 1.6000 4.2591 zn - k -0.3391 2.3200 4.9791 cu - mn -8.8791 -6.2200 -3.5609 *** cu - zn -4.2591 -1.6000 1.0591 cu - k -1.9391 0.7200 3.3791 k - mn -9.5991 -6.9400 -4.2809 *** k - zn -4.9791 -2.3200 0.3391 k - cu -3.3791 -0.7200 1.9391

  16. Dunnett's T tests for variable: MASS NOTE: This tests controls the type I experimentwise error for comparisons of all treatments against a control. Alpha= 0.05 Confidence= 0.95 df= 16 MSE= 1.81925 Critical Value of Dunnett's T= 2.592 Minimum Significant Difference= 2.2115 Comparisons significant at the 0.05 level are indicated by '***'. Simultaneous Simultaneous Lower Difference Upper TREAT Confidence Between Confidence Comparison Limit Means Limit mn - k 4.7285 6.9400 9.1515 *** zn - k 0.1085 2.3200 4.5315 *** cu - k -1.4915 0.7200 2.9315

  17. Type I Tukey’s test is recommended as the best! Type II Duncan’s test exaggarates the risk of Type I errors Scheffe’s test exaggarates the risk of Type II errrors Comparison between multiple tests

  18. PROC GLM; CLASS treat; MODEL mass = treat /SOLUTION; /* SOLUTION gives the estimated parameter values */ /* Test for differences between different levels of treatment */ MEANS treat / BON DUNCAN SCHEFFE TUKEY lines; RUN;

  19. General Linear Models Procedure Duncan's Multiple Range Test for variable: MASS NOTE: This test controls the type I comparisonwise error rate, not the experimentwise error rate Alpha= 0.05 df= 16 MSE= 1.81925 Number of Means 2 3 4 Critical Range 1.808 1.896 1.951 Means with the same letter are not significantly different. Duncan Grouping Mean N TREAT A 64.3000 5 mn B 59.6800 5 zn B C B 58.0800 5 cu C C 57.3600 5 k

  20. General Linear Models Procedure Tukey's Studentized Range (HSD) Test for variable: MASS NOTE: This test controls the type I experimentwise error rate, but generally has a higher type II error rate than REGWQ. Alpha= 0.05 df= 16 MSE= 1.81925 Critical Value of Studentized Range= 4.046 Minimum Significant Difference= 2.4406 Means with the same letter are not significantly different. Tukey Grouping Mean N TREAT A 64.3000 5 mn B 59.6800 5 zn B B 58.0800 5 cu B B 57.3600 5 k

  21. General Linear Models Procedure Bonferroni (Dunn) T tests for variable: MASS NOTE: This test controls the type I experimentwise error rate, but generally has a higher type II error rate than REGWQ. Alpha= 0.05 df= 16 MSE= 1.81925 Critical Value of T= 3.01 Minimum Significant Difference= 2.5663 Means with the same letter are not significantly different. Bon Grouping Mean N TREAT A 64.3000 5 mn B 59.6800 5 zn B B 58.0800 5 cu B B 57.3600 5 k

  22. General Linear Models Procedure Scheffe's test for variable: MASS NOTE: This test controls the type I experimentwise error rate but generally has a higher type II error rate than REGWF for all pairwise comparisons Alpha= 0.05 df= 16 MSE= 1.81925 Critical Value of F= 3.23887 Minimum Significant Difference= 2.6591 Means with the same letter are not significantly different. Scheffe Grouping Mean N TREAT A 64.3000 5 mn B 59.6800 5 zn B B 58.0800 5 cu B B 57.3600 5 k

  23. PROC GLM; CLASS treat; MODEL mass = treat /SOLUTION; /* SOLUTION gives the estimated parameter values */ /* In unbalanced (and balanced) designs LSMEANS can be used: */ LSMEANS treat /TDIF PDIFF; RUN;

  24. The GLM Procedure Least Squares Means LSMEAN treat mass LSMEAN Number cu 58.0800000 1 k 57.3600000 2 mn 64.3000000 3 zn 59.6800000 4 Least Squares Means for Effect treat t for H0: LSMean(i)=LSMean(j) / Pr > |t| Dependent Variable: mass i/j 1 2 3 4 1 0.844027 -7.29145 -1.87562 0.4111 <.0001 0.0791 2 -0.84403 -8.13548 -2.71964 0.4111 <.0001 0.0151 3 7.291455 8.135482 5.41584 <.0001 <.0001 <.0001 4 1.875615 2.719642 -5.41584 0.0791 0.0151 <.0001 NOTE: To ensure overall protection level, only probabilities associated with pre-planned comparisons should be used. Er denne P-værdi signifikant?

  25. Signifikante P-værdier Den sekventielle Bonferroni-test Den sekventielle Bonferroni-test er mindre konservativ end den ordinære Bonferroni-test. Procedure: Først ordnes de k P-værdier i voksende rækkefølge. Lad P(i) betegne den i’te P-værdi efter at værdierne er blevet ordnet i voksende rækkefølge. Herefter beregnes hvor α er det signifikansniveau, der benyttes, hvis der kun var en enkelt P-værdi (sædvanligvis 0.05). Hvis P(i) < α(i) er den i’te P-værdi signifikant.

  26. iid = independently and identically distributed Model assumptions and model control • All GLM’s are based on the assumption that (1) ε is independently distributed (2) ε is normally distributed with the mean = 0 (3) The variance of ε (denoted σ2) is the same for all values of the independent variable(s) (variance homogeneity) (4) Mathematically this is written as ε is iid ND(0; σ2)

  27. Transformation of data • Transformation of data serves two purposes • To remove variance heteroscedasticity • To make data more normal Usually a transformation meets both purposes, but if this is not possible, variance homoscedasticity is regarded as the most important, especially if sample sizes are large

  28. How to choose the appropriate transformation?

  29. We have to find a value of p, so that the transformed values of y (denoted y*) y* = yp meet the condition of being normally distributed and with a variance that is independent of y*. A useful method to find p is to fit Taylor’s power law to data

  30. Taylor’s power law It can be shown that p = 1- b/2 is the appropriate transformation we search for

  31. If y is a proportion, i.e. 0 <= y <= 1, an appropriate transformation is often

  32. T. urticae: log s2 = 1.303 + 1.943 log x r2 = 0.994 y* = log(y+1) P. persimilis: log s2 = 1.193 + 1.900 log x r2 = 0.992 y* = log(y+1)

  33. b = birth rate/capita d = death rate/capita r = net growth rate/capita Instantaneous growth rate B = birth rate D = death rate N = population size at time t ΔN = change in N during Δt δ = noise associated with deaths ε = noise associated with births Exponential growth Deterministic model: Stochastic model: The number of births during a time interval follows a Poisson distribution with mean BΔt The probability that an individual dies during Δt is θ = DΔt/N The number of deaths during a time interval is binomially distributed with parameters (θ, N)

  34. Type I, II, III and IV SS Example: Mites in stored grain influenced by temperature (T) and humidity (H)

  35. DATA mites; INFILE'h:\lin-mod\besvar\opg1-1.prn' FIRSTOBS=2; INPUT pos $ depth T H Mites; /* pos = position in store */ /* depth = depth in m */ /* T = Temperature of grain */ /* H = Humidity of grain */ /* Mites = number of mites in sampling unit */ logMites = log10(Mites+1);/* log transformation of Mites */ T2 = T**2; /* square temperature */ H2 = H**2; /* square humidity */ TH = T*H; /* product of temperature and humidity */ PROC GLM; CLASS pos; MODEL logMites = T T2 H H2 TH /SOLUTION SS1 SS3; RUN;

  36. General Linear Models Procedure Dependent Variable: LOGMITES Source DF Sum of Squares Mean Square F Value Pr > F Model 5 2.72839285 0.54567857 2.94 0.0265 Error 33 6.12429305 0.18558464 Corrected Total 38 8.85268590 R-Square C.V. Root MSE LOGMITES Mean 0.308199 85.66578 0.43079535 0.50287914 T for H0: Pr > |T| Std Error of Parameter Estimate Parameter=0 Estimate INTERCEPT 28.03994955 0.54 0.5902 51.56270293 T -0.86682324 -1.27 0.2147 0.68517409 T2 0.02333784 2.19 0.0358 0.01066368 H -3.52741058 -0.50 0.6235 7.11853025 H2 0.12548846 0.51 0.6161 0.24789107 TH 0.02315214 0.43 0.6702 0.05388643

  37. General Linear Models Procedure Dependent Variable: LOGMITES Source DF Sum of Squares Mean Square F Value Pr > F Model 5 2.72839285 0.54567857 2.94 0.0265 Error 33 6.12429305 0.18558464 Corrected Total 38 8.85268590 R-Square C.V. Root MSE LOGMITES Mean 0.308199 85.66578 0.43079535 0.50287914 Source DF Type I SS Mean Square F Value Pr > F T 1 0.22115656 0.22115656 1.19 0.2829 T2 1 1.38171889 1.38171889 7.45 0.0101 H 1 1.03546840 1.03546840 5.58 0.0242 H2 1 0.05579073 0.05579073 0.30 0.5872 TH 1 0.03425827 0.03425827 0.18 0.6702 Source DF Type III SS Mean Square F Value Pr > F T 1 0.29703065 0.29703065 1.60 0.2147 T2 1 0.88889243 0.88889243 4.79 0.0358 H 1 0.04556941 0.04556941 0.25 0.6235 H2 1 0.04755847 0.04755847 0.26 0.6161 TH 1 0.03425827 0.03425827 0.18 0.6702

  38. Example: β3 SS I is used to compare the model: with SS III is used to compare the model with

  39. General Linear Models Procedure Dependent Variable: LOGMITES Source DF Sum of Squares Mean Square F Value Pr > F Model 5 2.72839285 0.54567857 2.94 0.0265 Error 33 6.12429305 0.18558464 Corrected Total 38 8.85268590 R-Square C.V. Root MSE LOGMITES Mean 0.308199 85.66578 0.43079535 0.50287914 Source DF Type I SS Mean Square F Value Pr > F T 1 0.22115656 0.22115656 1.19 0.2829 T2 1 1.38171889 1.38171889 7.45 0.0101 H 1 1.03546840 1.03546840 5.58 0.0242 H2 1 0.05579073 0.05579073 0.30 0.5872 TH 1 0.03425827 0.03425827 0.18 0.6702 Source DF Type III SS Mean Square F Value Pr > F T 1 0.29703065 0.29703065 1.60 0.2147 T2 1 0.88889243 0.88889243 4.79 0.0358 H 1 0.04556941 0.04556941 0.25 0.6235 H2 1 0.04755847 0.04755847 0.26 0.6161 TH 1 0.03425827 0.03425827 0.18 0.6702 H is significant if it is added after T and T2 H is not significant if it is added after T, T2, H2, and TH

  40. How do we choose the best model?

  41. DATA mites; INFILE'h:\lin-mod\besvar\opg1-1.prn' FIRSTOBS=2; INPUT pos $ depth T H Mites; /* pos = position in store */ /* depth = depth in m */ /* T = Temperature of grain */ /* H = Humidity of grain */ /* Mites = number of mites in sampling unit */ logMites = log10(Mites+1);/* log transformation of Mites */ T2 = T**2; /* square temperature */ H2 = H**2; /* square humidity */ TH = T*H; /* product of temperature and humidity */ PROC STEPWISE; MODEL logMites = T T2 H H2 TH /MAXR; RUN;

  42. Maximum R-square Improvement for Dependent Variable LOGMITES Step 1 Variable H2 Entered R-square = 0.11939020 C(p) = 7.00650467 DF Sum of Squares Mean Square F Prob>F Regression 1 1.05692394 1.05692394 5.020.0312 Error 37 7.79576196 0.21069627 Total 38 8.85268590 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP -2.00838948 1.12364950 0.67311767 3.19 0.0821 H2 0.01218833 0.00544190 1.05692394 5.02 0.0312 Bounds on condition number: 1, 1 ----------------------------------------------------------------------------------- The above model is the best 1-variable model found.

  43. Step 2 Variable T Entered R-square = 0.14111324 C(p) = 7.97028096 DF Sum of Squares Mean Square F Prob>F Regression 2 1.24923115 0.62461557 2.960.0647 Error 36 7.60345475 0.21120708 Total 38 8.85268590 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP -1.75010488 1.15711557 0.48315129 2.29 0.1391 T -0.02071757 0.02171178 0.19230720 0.91 0.3463 H2 0.01202664 0.00545113 1.02807459 4.87 0.0338 Bounds on condition number: 1.000967, 4.003869

  44. Step 3 Variable H2 Removed R-square = 0.18352305 C(p) = 5.94726448 Variable TH Entered DF Sum of Squares Mean Square F Prob>F Regression 2 1.62467193 0.81233596 4.05 0.0260 Error 36 7.22801397 0.20077817 Total 38 8.85268590 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP 0.72634839 0.24079746 1.82684898 9.10 0.0047 T -0.52367627 0.19084468 1.51175757 7.53 0.0094 TH 0.03507940 0.01326789 1.40351537 6.99 0.0121 Bounds on condition number: 81.35444, 325.4178 The above model is the best 2-variable model found.

  45. Step 4 Variable T2 Entered R-square = 0.30260874 C(p) = 2.26668560 DF Sum of Squares Mean Square F Prob>F Regression 3 2.67890010 0.89296670 5.06 0.0051 Error 35 6.17378580 0.17639388 Total 38 8.85268590 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP 3.12310821 1.00603496 1.69993153 9.64 0.0038 T -0.94651154 0.24882419 2.55240831 14.47 0.0005 T2 0.02187819 0.00894923 1.05422818 5.98 0.0197 TH 0.03099168 0.01254804 1.07602465 6.10 0.0185 Bounds on condition number: 157.4125, 1019.366 ----------------------------------------------------------------------------------- The above model is the best 3-variable model found.

  46. Step 5 Variable H2 Entered R-square = 0.30305192 C(p) = 4.24554515 DF Sum of Squares Mean Square F Prob>F Regression 4 2.68282344 0.67070586 3.70 0.0133 Error 34 6.16986246 0.18146654 Total 38 8.85268590 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP 2.56528922 3.92853717 0.07737622 0.43 0.5182 T -0.85413336 0.67705608 0.28880097 1.59 0.2157 T2 0.02264025 0.01045241 0.85138509 4.69 0.0374 H2 0.00311049 0.02115432 0.00392334 0.02 0.8840 TH 0.02338380 0.05328321 0.03494992 0.19 0.6635 Bounds on condition number: 1451.704, 10936.61

  47. Step 6 Variable TH Removed R-square = 0.30432962 C(p) = 4.18459648 Variable H Entered DF Sum of Squares Mean Square F Prob>F Regression 4 2.69413458 0.67353364 3.72 0.0129 Error 34 6.15855132 0.18113386 Total 38 8.85268590 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP 26.64541542 50.83962556 0.04975537 0.27 0.6036 T -0.58573429 0.20112070 1.53634027 8.48 0.0063 T2 0.02565523 0.00908804 1.44347920 7.97 0.0079 H -3.55394542 7.03238763 0.04626106 0.26 0.6166 H2 0.13533410 0.24385185 0.05579073 0.31 0.5825 Bounds on condition number: 2335.775, 19486.21 ----------------------------------------------------------------------------------- The above model is the best 4-variable model found.

  48. Step 7 Variable TH Entered R-square = 0.30819944 C(p) = 6.00000000 DF Sum of Squares Mean Square F Prob>F Regression 5 2.72839285 0.54567857 2.94 0.0265 Error 33 6.12429305 0.18558464 Total 38 8.85268590 Parameter Standard Type II Variable Estimate Error Sum of Squares F Prob>F INTERCEP 28.03994954 51.56270293 0.05488139 0.30 0.5902 T -0.86682324 0.68517409 0.29703065 1.60 0.2147 T2 0.02333784 0.01066368 0.88889243 4.79 0.0358 H -3.52741058 7.11853025 0.04556941 0.25 0.6235 H2 0.12548846 0.24789107 0.04755847 0.26 0.6161 TH 0.02315214 0.05388643 0.03425827 0.18 0.6702 Bounds on condition number: 2355.783, 37061.88 ----------------------------------------------------------------------------------- The above model is the best 5-variable model found. No further improvement in R-square is possible.

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